Tomographic imaging methods are distinguished in that it is possible to examine internal structures of a patient or of a test object without having in the process to operate on the patient or to damage the test object. One possible type of tomographic imaging is to record a number of projections of the object to be examined from various angles. A 3D description (a virtual 3D model which can be represented in the computer) of the object can be calculated from these projections.
A standard method for this calculation is “filtered back projection”, FBP, (described in “Buzug: Einführung in die Computertomographie [Introduction to computed tomography]. 1st edition 2004. Springer. ISBN 3-540-20808-9” and in “Kak, Stanley: Principles of Computerized Tomographic Imaging. 1987, IEEE Press. ISBN 0-87942-198-3). What is involved here is an analytical method in which the projections are filtered and back projected onto the image.
However, this method can be used to reconstruct only points for which beams are present from an angular range of at least 180°. If this requirement is not met, strong visible artifacts result in the reconstructed image. This problem occurs with particular severity in medical computed tomography, where it is possible as a rule to reconstruct exactly all the points inside a circle, but not points outside the circle, for which reason organs or organ parts (for example arms, pelvis etc.) located in the outer region of the circle are affected by strong artifacts. For such points in the outer region, the detector is too small (or the object too large), and so these can no longer be projected thereon from all directions. This situation is particularly problematic when the aim is to determine the contour of the object to be reconstructed. Even this is no longer directly possible in such a case.
Currently, the standard solution to this problem is to estimate the missing projection beams and thereby to supplement the respective projections to the extent that the entire object is imaged on all projections. Filtered back projection (described in “S. Schaller, O. Sembritzki, T. Beyer, T. Fuchs, M. Kachelriess, T. Flohr, “An Algorithm for Virtual Extension of the CT Field of Measurement for Application in Combined PET/CT Scanners”, RSNA 2002 Vortrag”) is subsequently applied to the projections thus supplemented. This method is not exact and is very susceptible to error.
Iterative methods (for example the “algebraic reconstruction technique ART” or the “simultaneous algebraic reconstruction technique SART”) are also proposed for such problematic reconstruction methods in the same literature. Iterative methods are based on the principle that the measured projections are compared with the projections calculated from the object already reconstructed, and the error is subsequently applied for the correction of the image of the object. For example, in this case the image in the nth iteration Xn is calculated as follows with the aid of the update equation:Xn=Xn-1+{circumflex over (R)}(Y−{circumflex over (P)}Xn-1)  (1)
The iteration is begun by suitably initializing the starting image X0 (for example with the zero image by filling all the object values with zeros for the present). Here, {circumflex over (P)} in the above equation (1) represents the system matrix with the aid of which the respective projections are calculated from the scanned object image using knowledge of the scanning geometry. {circumflex over (R)} is the back projection operator.
So-called secondary conditions, for example in the form of a density distribution of the basic object material, can still advantageously be introduced into these iterative methods during the reconstruction. Specifically, it is described in the literature that such iterative methods can be used or applied successfully for the reconstruction in the case of the problem presented here when, in particular, it is known that the object consists only of one material and the formulation of the problem can be reduced to whether material is present or not at a specific point (“Robert N., F. Peyrin and M. J. Yaffe, Binary vascular reconstruction from a limited number of cone beam projections, Med. Phys. 21 (1994), 1839-1851”).
In this case, the formulation of the problem in which only a single type of material must be reconstructed is denoted as “discrete tomography”. In the latter, a threshold value analysis is applied to the image during the reconstruction, generally after each iteration, in order to take the decision on “material yes or no”. It is very problematic and therefore disadvantageous in the case of a discrete tomographic threshold value analysis that, on the one hand, a pixel is allocated only to a specific class (material/no material), while on the other hand this allocation can be corrected again—if at all—only by a very high number of iterations.